Given a smooth projective variety X with a smooth nef divisor D and a positive integer r, we construct an I-function, an explicit slice of Givental's Lagrangian cone, for Gromov--Witten theory of the root stack X_{D,r}. As an application, we also obtain an I-function for relative Gromov--Witten theory following the relation between relative and orbifold Gromov--Witten invariants.

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In this paper, we study the behavior of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa’s conjecture: Ricci-flat Calabi–Yau manifolds related by extremal transitions and flops can be connected by a path consisting of continuous families of Ricci-flat Calabi–Yau manifolds and a compact metric space in the Gromov–Hausdorff topology. In an essential step of the proof of our main result, the convergence of Ricci-flat K¨ahler metrics on Calabi–Yau manifolds along a smoothing is established, which can be of independent interest.

We view regularized learning of a function in a Banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel Banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization equation for the minimizer is also established.

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